Banach-valued Holomorphic Functions on the Maximal Ideal Space of H^∞

2011/03/11 by Alexander Brudnyi, Brudnyi, Alexander
Mathematics · #30D55 #30H05 #Advanced Banach Space Theory #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1103.2347

openalex publication_date 2011/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study Banach-valued holomorphic functions defined on open subsets of the maximal ideal space of the Banach algebra H^∞ of bounded holomorphic functions on the unit disk D⊂ C with pointwise multiplication and supremum norm. In particular, we establish vanishing cohomology for sheaves of germs of such functions and, solving a Banach-valued corona problem for H^∞, prove that the maximal ideal space of the algebra H_{\rm comp}^∞ (A) of holomorphic functions on \Di with relatively compact images in a commutative unital complex Banach algebra A is homeomorphic to the direct product of maximal ideal spaces of H^∞ and A.

Citations

• On covering dimension and inverse limits of compact spaces

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• Homage à A. M. Gleason et I. J. Schark